5.2: Circle Equations | Geometry Honors

How it works

A circle is the set of points a fixed distance from a center. Read the center after writing the equation in squared form. Complete the square in both variables to convert an expanded equation.

(x-h)^2+(y-k)^2=r^2

Worked examples

Example 1

Find the center and radius of $\displaystyle (x-2)^2+(y+3)^2=25$ .

The center is $\displaystyle (2,-3)$ and the radius is $\displaystyle 5$ .

Example 2

Convert $\displaystyle x^2+y^2-4x+6y-12=0$ to standard form.

Complete squares: $\displaystyle (x-2)^2+(y+3)^2=25$ . The added constants are $\displaystyle 4$ and $\displaystyle 9$ .

Problems

Problem 5.2.1

Write the equation of a circle with center $\displaystyle (2,1)$ and radius $\displaystyle 3$ .

Circle centered at 2,1 with radius 3 and a marked horizontal radius. — The center and radius determine the standard-form equation.

Show worked solution

Use the center-radius form: $\displaystyle (x-2)^2+(y-1)^2=9$ .

Circle centered at 2,1 with radius 3 and a marked horizontal radius. The solution adds labeled relationships. — Subtract center coordinates inside the squares; use radius squared on the right.

Problem 5.2.2

Does $\displaystyle (5,1)$ lie on the circle $\displaystyle (x-2)^2+(y-1)^2=9$ ?

Show worked solution

Yes. Substitution gives $\displaystyle 3^2+0^2=9$ .

Problem 5.2.3

A diameter has endpoints $\displaystyle (-2,1)$ and $\displaystyle (4,1)$ . Write the circle equation.

Show worked solution

The midpoint is $\displaystyle (1,1)$ , and half the diameter is $\displaystyle 3$ . Thus $\displaystyle (x-1)^2+(y-1)^2=9$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 5.2.1

Problem 5.2.2

Problem 5.2.3

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